Synonyms: IV4 assumption, fourth MR assumption

This is one of a set of additional (to the core) MR assumptions that are required for a well-defined causal parameter.

Two types of monotonicity exist when making causal inference in instrumental variable (IV) (including MR studies). Firstly, deterministic monotonicity is often referred to simply as “monotonicity”, which assumes a monotonic relationship between the IV and exposure. In other words, the association between the genetic IV and exposure should be either positive or negative across everyone in the sample. For example, a genetic IV should not increase the exposure in some people and decrease it in others. There can, however, be individuals within the sample where there is no effect of the genetic IV on the exposure. If this monotonicity assumption holds, then the IV estimate is consistent with the average causal effect (ACE) among compliers, where compliers are the subgroup of the sample affected by the genetic IV. However, defining who this subgroup might be is unclear. Alternatively, stochastic monotonicity, which is a relaxation of deterministic monotonicity, only requires that a monotonic increasing association between the IV and the exposure to exist conditionally on a set of covariates (which may or may not be measured). If this holds, then the IV estimate is consistent with a weighted average of treatment effects, such that more weight is given to treatment effects among subgroups where the effect of the IV on the exposure is greater. This assumption applies to both binary and continuous exposures but allows identifying an estimate that is less useful than the ACE among compliers. It is difficult in practice to know how important violation of these additional assumptions are in MR studies. Large genome-wide association study (GWAS) collaborations increasingly combine results from many studies (e.g., those across multiple European samples) and show consistency of association between genetic variants and traits across these studies at genome-wide significance (i.e., those variants used in most MR studies). This therefore suggests that homogeneity and monotonicity in effect estimates may exist for several MR studies in European populations. Non-parametric methods that provide bounds of causal effect estimates requiring only the core MR assumptions to be met may be applicable for some MR studies where violations of any of these additional assumptions is possible.

## References

- Sheehan NA, Didelez V. Epidemiology, genetic epidemiology and Mendelian randomisation: more need than ever to attend to detail. Human Genetics 2019; 139: 121-136.
- Swanson SA, Hernán MA. The challenging interpretation of instrumental variable estimates under monotonicity. International Journal of Epidemiology 2017; 47: 1289-1297.
- Small DS, Tan Z, Ramashai RR, Lorch SA, Brookhart MA. Instrumental Variable Estimation with a Stochastic Monotonicity Assumption. Statistical Science 2017; 32: 561-579.

## Other terms in 'Sources of bias and limitations in MR':

- Assortative mating
- Canalization
- Collider
- Collider bias
- Conditional F-statistic for multiple exposures
- Confounding
- Exclusion restriction assumption
- F-statistic
- Harmonization (in two-sample MR)
- Homogeneity Assumption
- Horizontal Pleiotropy
- Independence assumption
- INstrument Strength Independent of Direct Effect (InSIDE) assumption
- Intergenerational (or dynastic) effects
- MR for testing critical or sensitive periods
- MR for testing developmental origins
- No effect modification assumption
- NO Measurement Error (NOME) assumption
- Non-linear MR
- Non-overlapping samples (in two-sample MR)
- Overfitting
- Pleiotropy
- Population stratification
- R-squared
- Regression dilution bias (attenuation by errors)
- Relevance assumption
- Reverse causality
- Same underlying population (in two-sample MR)
- Statistical power and efficiency
- Vertical pleiotropy
- Weak instrument bias
- Winner's curse